(2x+3)-(5x-7)/6x+11=-8/3
To solve the given equation: (2x+3)-(5x-7)/(6x+11)=-8/3, we will follow these steps:
Step 1: Simplify the expression
Step 2: Clear the fractions
Step 3: Simplify the equation
Step 4: Isolate the variable
Step 5: Solve for x
Let's go through each step in detail:
Step 1: Simplify the expression
Start by simplifying the expression on the left side of the equation:
(2x+3) - (5x-7)
To simplify, remove the parentheses and distribute the negative sign:
(2x + 3) - 5x + 7
Combine like terms:
2x - 5x + 3 + 7
Simplify further:
-3x + 10
Step 2: Clear the fractions
Multiply the entire equation by the common denominator of the fractions, which is (6x+11).
By multiplying the equation by (6x+11), we eliminate the denominators, simplifying the equation.
(6x+11)[(2x+3) - (5x-7)/(6x+11)] = (6x+11)(-8/3)
Simplifying the left side:
(6x+11)(2x + 3) - (6x+11)(5x-7)/(6x+11) = -8(6x+11)/3
On the left side:
(6x+11)(2x + 3) simplifies to 12x^2 + 36x + 22x + 33
And on the right side:
-8(6x+11)/3 simplifies to -16x - 88/3
Simplifying further:
12x^2 + 58x + 33 - (6x+11)(5x-7)/(6x+11) = -16x - 88/3
Step 3: Simplify the equation
Cancel out (6x+11) from the numerator and denominator on the left side of the equation:
12x^2 + 58x + 33 - (5x-7) = -16x - 88/3
Simplifying further:
12x^2 + 58x + 33 - 5x + 7 = -16x - 88/3
Combine like terms:
12x^2 + 53x + 40 = -16x - 88/3
Step 4: Isolate the variable
To isolate the variable, we will move all the terms containing x to one side of the equation.
Add 16x to both sides:
12x^2 + 53x + 40 + 16x = -88/3 + 16x
Combine like terms:
12x^2 + 69x + 40 = -88/3 + 16x
Next, move -88/3 to the left side:
12x^2 + 69x + 40 - 16x + 88/3 = 0
Combine like terms:
12x^2 + 53x + 40 + 88/3 = 0
Step 5: Solve for x
To solve the quadratic equation, you can use factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula, which states that for the quadratic equation ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 12, b = 53, and c = 120/3 (or 40).
Substitute the values into the quadratic formula:
x = [-(53) ± √((53)^2-4(12)(120/3))] / (2·12)
Simplifying further:
x = [-53 ± √(2809-1920)] / 24
x = [-53 ± √(889)] / 24
The square root of 889 is an irrational number, so we leave it in the simplified form.
Therefore, the solutions for x are:
x = (-53 + √(889)) / 24
x = (-53 - √(889)) / 24